INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1954
Pnissuw.
LB. P i a 50
1291
IY
Figure 12. Predicted and Experimental Compressibility Factor for n-Pentane with Molal Volume as the Dependent Variable Subscript e = value of property calculated using the Benedict equation of state LITERATURE CITED (1) Beattie, J. A,, Hadlock, C., and Poffenberger, N., J . Chem. Phys., 3, 93 (1935). (2) Beattie, J. A., Levine, S. W., and Douslin, D. R., J . Am. Chem. Sac., 73, 4431 (1951). (3) I b i d . , 74,4778 (1952). (4) Beattie, J. A,, Simard, G. L., and Su, G.-J., Ibid., 61,26 (1939). (5) Ibid., p. 924. (6) Benedict, M., Ibid., 59, 2224 (1937). (7) Benedict, M., Webb, G. B., and Rubin, L. C . , Chem. Eng. Progr., 47,419 (1951). ( 8 ) Ibid., p. 449. (9) Benedict, M., Webb, G. B., and Rubin, L. C., J . Chem. P h y s , 8 , 334 (1940). (10) Ibid., 10, 747 (1942). (11) Benedict, M., Webb, G. B., Rubin, L. C., and Friend, L., Chem. Eng. Progr., 47,571 (1951). (12) Brough, H. W., Schlinger, W. G., and Sage, B. H., IND. E N G . CHEM.,43, 2442 (1951).
2ow
4000 PICSURE.
eo00
LB P C R sa
eo00 IN
Figure 13. Predicted and Experimental Compressibility Factor for n-Pentane with Pressure as the Dependent Variable (13) Hartree. D. R., “Numerical Analysis,” London, Oxford University Press, 1952. (14) Kvalnes, M., and Gaddy, V. L., J . Am. Chem. Sac., 53, 394 (1931). (15) Michels, A,, and Nederbragt, G. W., Physiea, 3, 569 (1936). (16) Olds, R. H., Reamer, H. H., Sage, B. H., and Lacey, W. S . , IND.ENG.CHEM.,35, 922 (1943). (17) Ibid., 36, 282 (1944).
(18) Reamer, H. H., Olds, R. H., Sage, B. H , and Lacey, W. N., Ibid., 36, 956 (1944). (19) Sage, B. H., and Lacey, W. N., Ibid., 34,730 (1942). (20) Sage, B. H., and Lacey, W . N., “Thermodynamic Properties of the Lighter Hydrocarbons and Nitrogen,” New York, Amer-
ican Petroleum Institute, 1950. (21) Selleck, F. T., Opfell, J. B., and Sage, B. H., IND.ENG.CHEM., 45, 1350 (1953). (22) Selleck, F. T., and Sage, B. H., Washington, D. C., Am. Documentation Inst., Doc. 3914 (1953). REChIVED
for review January 14, 1954.
ACCEPTEDFebruary 3, 1954
Residual Viscosity of Paint Systems at Infinite Shear Velocity J
W. K. ASBECK‘ AND M. VAN LOO The Sherwin- Williams Go., 115th St. and Cottage Grove Ave., Chirago, Ill.
T
HE critical pigment volume concentration of a paint ( 4 )
depends on the dispersive capacity of the vehicle used. Application qualities such as relative ease of brushing and flow also vary according t o the solid/solid and solid/liquid interactions occurring in the paint. Such relationships have been the subject of numerous rheological studies. These studies have been carried out almost exclusively a t low and intermediate shear velocities. Few high shear velocity measurements above 1000 reciprocal seconds (1000 sec. -1) have been attempted, apparently chiefly because of the lack of suitable instrumentation. 1
Present address, Carbide & Carbon Chemicals Co., 5. Charleston, W. Va.
BAND VISCOMETER
The band viscometer has been described elsewhere ( 1 1 , 1 4 ) . It is a simple high shear apparatus in which a thin flexible band is pulled between two thermostatted jaws, carrying paint with i t from a small well in the jaws. The apparent viscosity may be calculated from the equation
dK = 2WHV
where q is the viscosity at the given shear velocity, d is the clearance between the jaws, K is the wei h t pulling the band, W is the width of the band, H is the heigft of the active surface of
1292
INDUSTRIAL AND ENGINEERING CHEMISTRY
the j a m , and T i is the velocity of fall. The shear velocity D is equal to the relative linear velocity of one shearing surface against the other, divided by the distance of separation, or
The shearing stress T is equal to the force exerted on each square centimeter of one of the shearing surfaces, or
Hence the relationship reduces to the simple Sewtonian equation
v = j j The smaller the separation of the shearing surfaces a t a given velocity, the higher is the shear velocity. The unique feature of t,he instrument is that the band is self-centering a t very small clearances such as 50-100 as a consequence of the hydrodynamic relationships which iwre described in the reference ( 1 4 ) . , This allows shear velocities of 4000 t o 6000 see.-' with practical linear velocities of the band. Although the band viscometer permits measurements a t relatively high shear velocities, it was desirable t o obtain still higher values. This was difficult to accomplish readily even lyith modification of the band viscometer, because there is a limit t o t,he linear velocity that is practical to impart to the band. Also, the presence of the pigment in the paint under test limits the minimum clearance betm*een band and jaws. The apparatus has an added inconvenience when relatively thin paints are being measured, because its manipulation becomes somewhat untidy due t o paint dripping through the slot. This, of course, does not occur with heavier paints and pastes, and the instrument may be used to advantage in measuring the rheological properties of euch materials. S ELF-CESTERING ROTATIONA L VI 9CO9I ETER
I n order to circumvent the limitations of the band viscometer, a n instrument v a s designed t h a t combines the advantages of a rotational viscometer with the self-cent,ering action of the band viscometer ( 2 ) . I t s novel feature is a bob suspended flexibly by means of two universal joints rather than supported by a rigid shaft. One joint is a t the approximate geometrical center of the bob, and the other is some dist'ance above it. As a result, of this flexible suspension the bob centers itself automatically at bhe small clearances of about 100 microns between the bob and cup due to forces Fimilar t o t'hose operat'ing in t'he band viscometer. Shear velocities up t,o 30,000 see.-' are feasible with t,his instrument. The self-centering rotational viscometer was built primarily to study the correlation between the ease of brushing of paint,s and their viscosity a t the shear velocities t h a t pertain during actual application by brushing. These shear velocities are in a range approximating 10,000 reciprocal seconds. Since brushing involves continuous introduction of fresh paint to the forces applied, b o wells were provided in the annuli of the viscometer to introduce discontinuity of the same nature. Details nere described in the original paper ( 2 ) . STRUCTLRE AT LOW SHEAR VELOCITIES
A majority of the studies of the rheological structure of pigmented systems has attempted to resolve this structure t o definable terms. Considerable progress has been made in the low shear portion of the rheological system. Thus the work of Bingham ( 5 )introduced the concept of yield value which Houwink ( I O ) amplified, while Reiner and Riwlin (12) placed the theoretical flow of systems possessing structure on a firm basis in the low shear range in the rotational viscometer. Green and Weltman (9) showed the existence of the thixotropic hysteresis curve for many pigment/binder systems. Despite the quantity of work carried out a t low shear velocities,
Vol. 46, No. 6
no encompassing reliable equations, either theoretical or cmpirical, have been derived t o define precisely the flow of pigmented systems, probably because a t least two iridcpendent structure reducing mechanisms are superimposed in each measurement. These are the reduction of struct'ure through increasing shear velocity and time a t a constant shear velocity. So far no one has separated one of these mechanisms from the other. Consequently, an unknown sum of these two factors applies in each viscosity measurement. Either or both of these effects should be eliminated in order t o reduce the complexity of the rheological solid,'solid and solid/liquid interactions. Simpler and more definable relationships might thus be evolved. INFIXITE SHEAR VELOCITY
Theoretically, all struct,ure of a rheological system would be eliminated if the system could be measured at an infinitcly great shear velocity. At this point, systems possessing varying s tructural characteristics could be eva.luatcd on a comparable basis. Under these circumstances only the viscosity of the suspending medium, the shape and size of the suspended particles, and the volume concentration of the particles should play major roles in determining the residual viscosity characteristics of the material. One of the unfortunate fact,ors involved in the concept of vm, the viscosity of a system a t infinite shear velocity, is that it can never be evaluated in a practical measurement. Although high shear velocity measurements, certainly as high as 20,000 sec. -1, can be carried out on pigment-binder systems: the top shear velocities for measurement,s under la,minar conditions of flow are rest,ricted to the range of the Reynold's number for that particular sample of material. Above this value, turbulence sets in, and the linear relationship between stress and strain for Tewt,onian liquids no longer applies. Consequently, t,hcre exists a definite shear velocity for each material measured on a specific viscometer, above which the measurement no longer follow the laws derived for lower shear velocities. As a result,, only a mathematical value for q m can be obtained. This is the equivalent, of the viscosity of a material if it could be measured a t infinite shear velocity and if the l a w of laminar flow still held a t this point. Despite the apparent indefiniteness of this t,heoretical concept, the value of q m has considerable significance from a practical standpoint. Goodeve and Whitfield (8)were among the first workers to attempt the straight line ext,rapolation from low shear velocity measurements to infinite shear velocity by extrapolating t,he graph of the reciprocal of t,he shear velocity to zero. They developed a theoretical equation
where ?I* is bhe apparent measured viscosity, the residual viscosity at, infinite shear velocity, f the coefficient of thixotropy, and D the shear velocity in question. By plotting v* against I/D, they reported to have obtained a straight line relationship v i t h intercept qm on the * axis and slope f. The special viscometer developed by Goodeve and TThitfield for these measurements possessed the unique feat,ure of having a cone shaped bob and a cone shaped cup. The shear velocity could be varied by lowering or raising the bob with respect t o the cup, t,hus changing t,lie clearance between them. Unfortunately, the measurement,s carried out with this instrument seldom exceeded shear velocities of several hundred reciprocal seconds. These shear velocities were evidently not sufficiently high t o establish conclusively the validity of their theoretical deduct,ions. Measurements carried out on paints with the band viscometer and the self-centering rotational viscometer u p t o shear velocities of about 5000 see. and 20,000 see. -I, respectively, indicate t h a t for paintlike pigment/binder systems the points very often fall increasingly farther below the theoretical straight line of Goodeve and Whitfield the higher the shear velocities measured.
INDUSTRIAL AND ENGINEERING CHEMISTRY
June 1954 Fiom theoretical considerations it is true for all systems containing a thixotropic type of structure, except for Bingham bodies. The latter can be represented by the simple equation
1293
is driven a t exactly the same speed in the opposite direction by means of the differential transmission, T,as long as the housing of the transmission is prevented from rotating. Under these circumstances, no torque is imparted to the cup, C, despite the fact that the speed of rotation of the bobs may be varied over a wide range, because the torque produced on the cup by both bobs is equal and opposite. By means of a second, constant speed motor M p , a definite small rotational velocity A D can be imparted to the housing of the transmission T . This imparts a speed to B, equal to ( D A D ) and to BZequal t o (--D A D ) . The torque imparted t o the cup will then be equivalent to the difference between the apparent viscosity of the material being measured, a t a shear velocity D1 = ( D A D ) plus the value a t D, = ( - D AD) or r2 - T I = A T . Because of the fact that
+
+
on the typical stress-strain diagram where 7 is the true viscosity of the system, 7 the shearing stress, D the shearing strain, and f, the yield value. Thc apparent viscosity ? * as measurc,d with a rotational viscometer is generally expressed as the cotangent of stress and strain for that particular shear velocity, or
The Goodeve and W h i t f i e l d equation is obtained by solving the above eauations simultaneously and eliminating T , 7 being Figure Schematic Design of Absolute Visequal to their vm. cometer Knfortunately, most paints do not conform to the relatively simple rheological characteristics prescribed by the Bingham body. B s a consequence the relationship does not hold for the majority of pigment/binder systems. Rather, a more or less decided curvature toward the origin is found in the viscosity curves when recorded on the stress-strain diagram. The slope of the tangent to this curve a t any shear velocity is the true viscosity of the system a t that shearing strain. Usually the viscosity of a system is expressed as the apparent viscosity, or the measured stress, T , required to produce a given shear velocity D, or vice versa. The true viscosity and the apparent viscosity of a system consequently are not the same, the one being obtainable from the other only through the construction of the stress-strain diagram and obtaining graphically the tangent t o the curve a t any shear velocity. This is too laborious for most applications and too long a process to find general usage.
+
+
'
AT
K D
and 2 A D can be kept constant throughout the experiment, despite the fact that D can be varied over wide ranges, the true viscosity at any shear velocity becomes proportional to A T , or: 7 = K A T , where K is the over-all instrument constant. Thus, the instrument performs all of the necessary calculations, and the true viscosity of a fluid may be determined and recorded directly as a function of any shear velocity the instrument is capable of producing. The apparatus is shown schematically to illustrate the principles involved. Slight changes can, of course, be made in the mechanism, so long as the general principles of its functioning are maintained. HIGH SHEAR INTERPOLATION OF
vm
The problem still remains to find a reliable method of obtaining the viscosity of a material a t infinite shear velocity from measurements a t finite shear velocities. The most convenient method to accomplish this appears to be by graphical extrapolation as suggested originally by Goodeve and Whitfield. Since the extrapolation of curves is unreliable, some function of T and 1 / D should be found that results in straight lines u-hen the experimental data are plotted. Data for either apparent viscosity or absolute viscosity can be employed for this purpose since the line defining the slope of 7 * becomes parallel to the tangent to the true 7 a t infinite shear velocities.
7
AB SO LUTlL VISCOMETER
An instrumentation problem must be faced when seeking to obtain a direct reading of the true viscosity of a system at a given shear velocity by a single measurement. The problem is to design an apparatus that will express its results in terms of the cotangent to the true viscosity curve of that system, or to measure
t 9)
100 40 10 5
2 SHEAR
AT AD
a t a given shear velocity. In principle this would require construction of an instrument with oppositely rotating cylindrical hobs of identical dimensions where one of the bobs can be given a n exactly definable excess rotational velocity. The difference in torque produced on the cup is then proportional to the true viscosity. Although this may appear to be a difficult mechanical problem, an instrument filling the above requirements could be constructed rather simply by use of the mechanism of a differential transmission. 9viscometer of this kind is illustrated schematically in Figure 1. A variablt: speed motor, M I , drives the lower bob, B,, through a direct shaft. The upper bob, Bz,which has exactly the same dimensions and clearances to the cup as B I ,
Figure 2.
I
I
1
.E
VELOCITY
.6 in
I
I
.4
SEC? x
.3
IO-^
.2
Viscosity Characteristics of House Paints
Apparently reliable coordinates for producing straight lines from high shear data are described in a previous paper ( 2 ) . A graph employing this method of plotting log ? * against 1/40 is shown in Figure 2 for high shear data obtained from testing a number of commercial house paints. Good linearities are obtained between 20,000 see.-' and 300 see.?. At shear velocities below 300 set.-' the function falls off rapidly, and the method cannot be employed. By extrapolating the value of 1/45 to zero a theoretical value of the viscosity of the system at infinite shear velocity is obtained. The equation for these lines is
INDUSTRIAL AND ENGINEERING CHEMISTRY
1294
where S is the slope of the lines and is a function of the structure of the rheological system. This relationship is valid only for solid!liquid systems where the liquid does not contain a major degree of structure in comparison with the total structure of the rheological system. I n materials such as pigmented latex systems where the liquid phase may be very shear-sensitive due to the use of thickening agents and stabilizing agents, a curve usually results when the experimental points are plotted on the above coordinates. In such systems two independent, shear-sensitive mechanisms are superimposed which result in viscosity relationships that’ a,re different from those found in conventional paints. For the latter, the relationships suggcsted seem t o hold rather well. Only a few conventional paint systems do not give a straight line relationship on the above coordinates when measured a t high shear velocities, and these-like some of the new interior flat paints-contain vehicles which are not Xervtonian in nature. Since the above functions are purely empirical there is no reason to assume, however, that other functions would not be equally satisfact,ory. The validity of these functions can be substa,ntiated by practical experimentation. PRACTIC4L TvfEAVING OF 7.. qm has been defined as that point in a rheological system nhere all structure of the system has been broken down. Then if two systems Fan be found that have different structure but exactly the same ultimate particle size diatribution; contain exactly the same concentration of these particles, and are of such a nature that the viscosity of the liquid phases is identical for both systems, the extension of the cxtrapolated viscosity plot
60
40 20 v)
w
u-l
10
5
8
>b-
8
2
2 1
.6 .6
4
100 40 10 5
2
I
.8
.4
.6
SHEAR VELOCITY In SEC? x
Figure 3.
E
2
.3
2
IO-^
Effect of Flow Agent on‘ritanium DioxideRaw Linseed Oil Systems
X number of attempts have been made in recent times to extend the equation first introduced by Einstein ( 7 ) to predict the viscosity of pigmentlbinder systems from a knowledge of‘ the viscosity of the suspending liquid and the volumetric concentration of the pigment. Einstein’s theoretically derived equation holds only for relatively low concentrat’ions of pigment. T’and ( I S ) has extended this theoret’ical.development to considerably higher concentrations, in the order of 35%, while more recently Brailey (6) has pointed out that additional correction factors are required t o extend these values still higher.
5
s
4
*
3
t
should intersect a t the same point a t infinite shear velocity. Although it may appear difficult to obtain two samples with the% specifications, the requirements actually can be met easily. A number of so-called flow agents are available to the paint industry that when added in small quantity to paints may either thin t,he paints or “body them up,” depending upon the material used to obtain the desired effect. Water has a decided bodying action on many paints. This is part’icularly true of t,itiLniuin dioxide pigments dispersed in raw linseed oil ( I ), If the pigrric3iit is first dried carefully, dispersions of this material in raxy linaxd oil-thinner vehicles are relatively unagglomerated. The addit,ion of small quantities of water to this system producer: considerable agglomeration with attendant rheological structure. 9 series of paints was made up containing 5, 10, 15, 20, 25, and 30% titanium dioxide pigment by volume in a vehiclc containing 9670 raw linseed oil and 4% mineral spirit,s, using previously dried pigment. After grinding in a high speed miniaturc ball mill (S),the individual samples were split and 2% water addcd t,o one series and allowed to agc after shaking. The vivositic.s were determined on the high shear rotational viscometer. Thv results are shown in Figure 3. The curves for the various pigment concentrations intersect the ? * axis a t about the aan-e point, indicating the validity of the assumptions made. This relationship is not restricted to titanium dioxide-raw linseed oil syvtcms. Similar results are obtained when small quantities of natcr arc added to some commercial multipigment paint systems. The rheogram of a commercial multipigment, paint sxstcm is shown in Figure 4. I n this case 114, 1/2, and 1% of a flow agent were added to increase the flow of the original paint,. The slope of the line is lowered someiThat by 1/~70 of the agent, while the curves for 1/2 and 1 % arc lowered still more and practically coincide, indicating that 1 / p % of the material was probably enough to obtain maximum effect. Again all four lines seem to converyci. The point I ) - , the residual viscosity a t infinite shear vclocity is, then, a fundamental constant for each rheological system, and at infinite shear velocities is independent of the structure of that, system. VAND EQUATION
.s a
7101. 46, No. 6
a too e
::
B
PERCENT 112 P E R C E N T I PERCENT
114
C D
t
I l l (0
1 1
100 40 IO 5
I
2 SHEAR
Figure 4.
1
I
1
VELoctrr
I
I
.6
8 In
4
SEC-’
I
.3
100
2
0
30
‘Ioil
10-3
Effect of Flow Agent on Commercial House Paint
PO
10
Figure 5 .
Ratio of q m / m for Titanium DioxideBodied Linseed Oil System
INDUSTRIAL AND ENGINEERING CHEMISTRY
lune 1954
One of the early Vand equations states that 7 = 70(1
-c
- pc*)-k
wheie q is the viscosity of the system, 70 is the viscosity of the suspending medium, c is the volumetric concentration of the pigment, and q and k are constants of the system; q characterizes the amount of liquid immobilized around a colliding pair of particles a t any instance, and k is the shape factor of the particles. Although a number of studies have been carried out to substantiate these theoretical equations, only relatively large particles, such as glass spheres of about 100 p, have given successful results. In paint systems the correlations between theory and practice have been relatively unreliable. Whereas the large glass spheres display only negligible or no structure, the structure of most paintlike systems is relatively high and unpredictable. Small quantities of agents can change the viscosity of these svstems considerably at low shear velocities. Thus very often two,paint systems that are actually identical in ultimate viscosity at q m will be considered to differ substantially a t finite shear velocities because of their different degrees of structure. Referring to Figures 2, 3, and 4, which viscosity shall be substituted in the Vand equation to designate the 7 of the system? Shall it be the one obtainable a t the low finite shear velocity of the conventional viscometer, or the one obtainable a t relatively high shear velocities with the band instrument or the self-centciing rotational device? The designated figures show clearly that some residual structure remains in most paints even a t the highest practical shear velocities obtainable, The Vand equations are derived for systems assumed to have no structure. Thus the hypothetical q , becomes a logical choice for testing the derived equations on a practical basis for systems containing structure. As a consequence, the Vand derivation should be amended to include a very specific definition for 7, namely, that viscosity where no structure remains, or q m . RATIO OF
1295
dispersion a t various temperatures and extrapolate to q m . I n this type of system there is no structure at qm. The pigment is independent of temperature with respect to size, shape, and number of particles. Therefore, the viscosity change of the vehicle due to its temperature dependence is the only variable. This type plot is shown in Figure 5. A titanium dioxide system differing from that depicted in Figure 3 was dispersed a t 45% concentration by three passes on a one-roll mill in bodied linseed oil. It was then diluted to concentrations of 5, 10, 20, and 30% pigment volume using the same bodied linseed oil. The values of q m are shown plotted against the viscosity of the oil as 70measured at 20°, 30",and 50" C. on the band viscometer. A good linear relationship exists between q m and 70 within the experimental error. The same titanium dioxide pigment was dispersed in raw linseed oil as described and measured a t 20" and 30' C. This system is plotted in Figure 6. Although the linearity is good, the slopes of the curves differ somewhat from those of Figure 5. The temperature coefficient of these pigment/binder systems is also the same as that of the vehicles in which the pigments are dispersed. If the temperature/viscosity relationship for the vehicles is plotted on the familiar log 9 versus 1/T diagram, a substantially straight line is obtained. The pigmented systems made with the same liquid also have the same temperature coefficient as the vehicles themselves. Figure 7 illustrates this for the same bodied linseed oil dispersion of titanium dioxide. Similar results are obtained for the raw oil dispersions. The slopes of the lines for the different volume concentrations of pigment dispersed in the raw linseed oil and bodied linseed oil as plotted in the Figures 5 and 6 aro not the same. Either the pigment has been broken down differently in the grinding proc-
TO ?la
The T'and equation also predicts that the combined influence of such factors as the pigment concentration, pigment shape factor, and immobilized liquid is proportional to the ratio of the viscosity of the medium and to the viscosity of the oil system in which it is suspended, or,
If a given pigment system could be kept identical in all respects except the viscosity of the suspending oil, a graph of the viscosity of the oil versus q m for the system should result in a straight line if the above derivation is to hold. The simplest technique for obtaining the above conditions is to measure the same pigment
TEMPERATURE
*C.
Figure 7. Temperature-Viscosity Effects
le
8
8s
e
4
2
0
.2
.3
.4
.5
.6
'IOil Figure 6. Ratio of vm/7o for Titanium DioxideRaw Linseed Oil System
PIGMENT CONCENTRATION in PERCENT
Figure 8.
Vand Equation for Titanium DioxideBodied Linseed Oil System
I N D U S T R I A L A N D EN G I N E E R I N G C H E M I S T R Y
1296 30
20
2
I
0
9
20
IO
PIGMENT
Figure 9.
45
30
CBMCENTRATIOM in PERCENT
Vand E q u a t i o n for T i t a n i u m DiosideRaw Linseed Oil S y s t e m
esses for the two systems, resulting in a different shape factor, or the immobilized layer is differcnt for the two systems, raw oil and bodied oil. Figures 8 and 9 indicate that both phenomena have taken place. These Figures depict the Vand equation for the two titanium dioxide systems plotted log
11
= -k log (1
- c - yc2)
110
where the ordinate has been plotted as c where c i3 expressed log
1
1-c
- qcz
Different values of y are required to produce straight lines, and these values approach zero for the ram‘ oil system and one for the bodied oil system. The shape factor 12, as expressed by the slope for these lines, also varies and is larger for the raw oil system. Thus, both the type of oil used and the method of grinding can have a major effect on the rheological behavior of paint systems as is substantiated by general paint experience. CONCLUSIOR-S
The field of rheology dealing with high and very high shear velocities, which has been neglected previously, reveals many interesting and simple relationships not discernible from studies confined to only low shear velocities. Data obtained with tm‘o
high shear instrument,s are employed to demonstrate the validity of the hypothetical term the residual viscosity a t infinite shear velocity in practical viscomet,ry. A method of determining this value by the empirical plotting of log Q*, the apparent viscosity, against t,he reciprocal square root of t,he shear velocity is shown. At the point where D = a, all rheological structure has been eliminated. By applicaticn of these concepts to the derived equation of 1-and, closer conformity between practical rheology of paint systems and the theoretical equation can be shown. In particular, a close linear relationship between t,he value of t,he viscosity at infinite shear velocity of a pigmentibinder system an$ the viscosity of the binder is demonstrated. More worlr IS required before a comprehensive prediction of the form factor k and the immobilization constant p for paint systems is possible. Hon-ever, these constants seem to depend both on the vehicle used as the binder and the grind given t’he system. The theory of a viscometer with which it is possible t o determine the true viscosity of a non-Sewtonian system direct’ly is presented. LITERATURE CITED
(1) Asbeck, W.K.. Hers, G. P., and Van Loo, M., presented before the Division of Paint, Tarnish, and Plastics Chemistry at the 120th Meeting, AEvfER1c.m CHEMICALSOCIETY, New York, 1951. (2) bsbeck, W ,K., Laiderman, D. D., and Van Loo, AI,, J . Colloid Sci., 7, 306 (1952). (3) ilsbeck, W.L., Laiderman, D. D.. and Van Loo, AI., O&C. Dig. Paint & Varnish Production Clubs, No. 326, 156 (1952). (4) Asbeck, W.IC., and Van Loo, M., ISD. ENG.CHEM.,41, 1470 (1949). (5) Bingham, E. C., Natl. Bur. Standards, Sei. Papers 278 (1916); Proc. Am. SOC.Testing ilfnferials,19, Pt. 2 (1919). (6) Brailey, R. H., presented before the Division of Paint, Varnish, and Plastics Chemistry at the 120th Meeting, AMERICAN CHEMICAL SOCIETY, Kew York, 1951. (7) Einstein, A., Ann. Physik, 19, 289 (1906); 34,591 (1911). (8) Goodeve, C. F., and Whitfield, G. W.. Trans. Faraday SOC.,34, 511 (1938). (9) Green, H., and Weltman, R. N., IND.ENG.CHmf., ANAL.ED., 15, 201 (1913): 18, 167 (1946). (10) Houwink, R., “Elasticity, Plasticity, and Structure of Matter,” Cambridge, Cambridge Cniversity Press, 1937. (11) Hull, H. H., J. Colloid Sci., 7, 316 (1962). (12) Reiner. M., and Riwlin, R., Kolloid-Z., 43, 1 (1927). (13) Tand, V., Xaature, 155, 364 (1945); J . Phys. & Colloid Chem., 52, 277 (1948). (14) Wachholtz, F., and Asbeck, W. IC.,Kolloid-Z., 93, 280 (1940) ; 94, 66 (1941). RECEIVED for review November 13, 1953. ACCEPTED February 27, 1954. Presented before the Division of Paint, Plastics, and Printing Ink Chemistry a t the 124th Meeting, AMERICAN CHEarIcAL SOCIETY, Chicago, Ill., 1953.
Prediction of
F1
Vol. 46, No. 6
ivity
e
J
EMPLOYING CONSTANT CATAEY ST REPLACEMENT RATE S. L. A3H)ERSEN’ AND R. H. 3IATTHIAS Electrochemicals D e p a r t m e n t , Research DiGision, iYiugara Falls L a b o r a t o q , E . I . du P o n t d e N e m o u r s & Co., ITLC., Yiagara Fulls, iY. Y.
I
T IS important to be able to predict the activity of catalyst
in a fluidized bed or stirred vessel under continuous replacement. Studying the catalyst activity for various rates of catalyst replacement is very time-consuming. If several rates of replacement are being considered, several experiments must be performed, each of which requires hundreds of hours to reach steady-state activity. A mathematical analysis is proposed to 1
Present address, Korth Carolina State College, Raleigh, S . C.
predict this steady-state activity for any catalyst replacement rate from laboratory experiments ueing similar equipment but with no replacement of the catalyst. Present and potential applications of fluidization as discussed by Sittig ( 3 ) include phthalic anhydride manufacture, oxidation of ethylene to ethylene oxide, and synthesis of vinyl chloride from acetylene and hydrogen chloride. The prediction equations derived in this paper may prove applicable to these processes. Limited use appears feasible in such fields as petroleum cracking.
